An approximation of one-dimensional nonlinear Kortweg de Vries equation of order nine

This research presents the approximate solution of nonlinear Korteweg-de Vries equation of order nine by a hybrid staggered one-dimensional Haar wavelet collocation method. In literature, the underlying equation is derived by generalizing the bilinear form of the standard nonlinear KdV equation. The highest order derivative is approximated by Haar series, whereas the lower order derivatives are attained by integration formula introduced by Chen and Hsiao in 1997. The findings are shown in the form of tables and a figure, demonstrating the proposed technique’s convergence, robustness, and ease of application in a small number of collocation points.


Introduction
Many problems that have arisen in different eras of science and engineering have been described using linear and nonlinear phenomena. While some of these issues can be solved immediately, a considerable number of them remain at the cutting edge of mathematical modelling. As a result, partial differential equations (PDEs) have become an important tool for describing such processes. A clear understanding of modelling is essential to deal with such situations [1]. The authors were driven to find analytic, semi analytic, or numerical solutions to these models after studying the nature of their solutions. When finding analytic solutions to these PDEs proved difficult to come by, the authors became interested in semi-analytic and numerical solutions. Numerous semi-analytic techniques, such as Adomian decomposition method (ADM), homotopy analysis method (HAM), and homotopy perturbation method, have been utilized to produce series solutions, however convergence of the series has been a challenge in these solutions which was solved by many semi-analytic techniques like Adomian decomposition method (ADM), homotopy analysis method (HAM), homotopy perturbation method (HPM) and modified variational iteration method (MVIM) [2,3].
Furthermore, these techniques have successfully handled a variety of linear and nonlinear events. Many authors focused on linearization for nonlinear problems, which did not change a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 depending upon the initial condition(IC): and the boundary conditions(BCs): where $ðx; tÞ is an unknown function of two variables x (space variable) and t (temporal variable), the domain is denoted by � O, whereas the boundary is expressed by @ � O. The term $ 9x is linear and the terms 45 The major objective of this article is to propose a robust numerical technique, to approximate nonlinear KdV equation of order nine, that yields computed results in a small number of collocation points(CPs).
The proposed method in this article is a part of Ph.D. thesis [32]. It is organized in the following manner: Section 2 explains the topics of multi-resolution analysis and functions of Haar wavelet. Section 3 prescribes the proposed numerical scheme for KdV equation of order nine. Section 4 shows the convergence of the proposed method. A numerical example and justifications are given in Section 5. Finally, in Section 6, some conclusions are drawn from the proposed research work.

Multi-resolution analysis
The better understanding of wavelet functions can be achieved by multi-resolution analysis (MRA, a sequence of spaces { € Ϝ j }), with the following properties: where € f is a square integrable function over the real line and L 2 (R) is a function space, if the space € Ϝ j is defined as: then after scaling and translation, the analysis of multi-resolution(MR) can be constructed for f € Ϝ j ; j 2 Zg (the sequence of spaces) that are described by Eqs (5) and (7) on using € h 1 ðxÞ.

Haar wavelet functions
where π, ρ and σ are constants and j indicates the HW level, o is resolution level where o = 2 j , j = 0, 1, . . ., J with J is the maximum resolution level, is the translation parameter and the index i is used for wavelet number.
The father wavelet has the following representation: All the members of square integrable functions' family described on the interval [0, 1] can have the following representation, that is in the form of summation of members of HW family: We identify Θ = 2 J and Γ = 2Θ, where J is defined before. The summation of HW functions is an approximation of € f ðxÞ defined on the interval [0, 1). The characteristics for integrals of Haar functions are introduced by (5), the first two integrals may be calculated as:

From Eq
Generally, the interpretation for the Haar integrals(HI) is given by where i = 2, 3, . . . and for i = 1, we have
phenomenon, while its series solution is a hyperbolic function. In physics, distortion in onedimensional(1-D) rippling is given by the presented equation, that involves shallow water waves, likewise, in the routine work, hyperbolic functions perform a momentous role as well.
The one-dimensional HWCM is applied on the Eq (1) and Tables 1-3 illustrate its numerical results. The Table 1 represents the point-wise absolute errors (PWAEs) (for t = 1, 3,5,7,9). It is noticeable that for x = 0, the PWAE is vanished, whereas increase in the other PWAE is observed, if we move along space. The Tables 2 and 3 represent MAEs and the rate of convergence (respectively) at different collocation points for various time step sizes (Δt = 10 −02 , 10 −03 , 10 −04 ) and the constant time level t = 1. The obtained MAEs are representing the precision of one-dimensional HWCM for different collocation points. The order of PWAEs is 10 −02 along with Δt = 10 −02 , b = 0.2, t = 1 and � k ¼ 0:35. Moreover, the order of MAEs is 10 −01 for � k ¼ 0:35 and t = 1. In the prescribed problem, by lowering the time step size, the progress in the precision of the presented technique is observed, while it (precision) is not increased by increasing the number of collocation points. To fix this problem, some restrictions will be implemented on this procedure. In addition, the

Conclusion
The nonlinear KdV has been approximated by one-dimensional HWCM. The given equation is discretized utilizing finite difference technique and the collocation procedure. The proposed scheme is implemented on KdV equation of order nine and a reasonable performance of the one-dimensional HWCM is observed from the computed results. Furthermore, it is noted, that reducing the size of time step results in an improvement in the precision of the presented technique, while the precision does not increase with increasing collocation points. However, in future, a few constraints are needed to impose on the proposed scheme, to obtain the required increase in precision.